![]() ![]() There is a one-to-one correspondence between the Mersenne primes and the even perfect numbers. So, 6 is a perfect number because the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6. Meanwhile, perfect numbers are natural numbers that equal the sum of their positive proper divisors, which are divisors excluding the number itself. The numbers p corresponding to Mersenne primes must themselves be prime, although not all primes p lead to Mersenne primes-for example, 2 11 − 1 = 2047 = 23 × 89. For example, 3 is a Mersenne prime as it is a prime number and is expressible as 2 2 − 1. ![]() Mersenne primes, named after the friar Marin Mersenne, are prime numbers that can be expressed as 2 p − 1 for some positive integer p. Mersenne primes and perfect numbers are two deeply interlinked types of natural numbers in number theory. Visualization of 6 as a perfect number Logarithmic graph of the number of digits of the largest known prime by year, nearly all of which have been Mersenne primes ![]()
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